When you breed a CT to anything other than CT, you get a bunch of shaggy rayed fry - a lot of people think that these are homely, so finding home will be really difficult. Also, your male has some serious conformation faults. I love the female, but breeding that male could lead to deformed fry. I'd say yes to the female, just find her a nicer mate.

It's not "what's wrong with the CT" . . . but CT x what ever fin type = messy, uneven web reduction in the first few generations. This cross will only produce a handful of "decent" looking bettas. You need to further selectively breed them to either get solid fins or actual CT.

Yes. . . unless you want to improve your CT. Lets say that your CT is a DeT and you want to make its caudal spread even further. Then you would have to cross it to a decent HM and work yourself through - selectively breeding them to get HMCT. If you breed them correctly, it should take you 3-4 generations of breeding.
Or say you want your CT to have bigger fins and at the same time wider spread. Then you'd have to breed it to a HMDT and selectively breed offspring. . . . . and so on